Poynting Vector question

Main Question or Discussion Point

Poynting Vector is by definition:

S = 1/u(E x B), where S points in the direction of the EM wave's motion.

In other words, for an EM wave moving from left to right, the electric field component always points up as the magnetic field component hits us in the face, and conversely, the electric field component always points down as the magnetic field components moves away from us into the page.

My question is, how was this constant relative position of the B field to the E field determined, i.e. was it experimentally determined, or is it a necessary mathematical consequence of other formulae? In other words, why is it E x B instead of B x E? Is this just the way EM fields are measured to be, or is their some logical/mathematical reason for it?

All responses appreciated.

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We know from Maxwell's equations in vacuum that the Laplacian(E) = UoEo*(d^2E/dt^2) and similarly for Laplacian(B) = UoEod^2B/dt.

These satisfy the wave equation. However, Maxwell's equations add constraints to the waves.

For instance, a wave traveling in the z driection Ez and Bz must equal zero to satisfy div(E) = div(B) = 0. (Waves are in a vacuum so div(E) = 0).

Also del X E also tells us that Bo = k/w(z X Eo) which is the relation you were discussing.

Oh and sorry about the messy notation, I don't know LaTex.